which is a Hamiltonian Dysthe equation for two-dimensional gravity waves on deep water with constant vorticity.
It describes modulated waves moving in the positive \(x\) -direction at group velocity
\(\Omega _0^{\prime } = \partial _{k} \Omega (k_0) = g/(2 \omega _0)\)  as indicated by the advection term.
The nonlocal term \(u |D_X| |u|^2\)  is a signature of the Dysthe equation, which reflects the presence of the wave-induced mean flow.
The coefficient \(\beta \)  is given by (REF ).
The coefficient of the nonlinear term \(|u|^2 u\)  above agrees with that in the NLS equation of [1]}
and in the Dysthe equation of [2]} (see (REF ) below) up to scaling factors consistent with
the difference in definition for the wave envelope described in the various models.
